In the paper the axisymmetric problem of elasticity theory is studied for the radially inhomogeneous sphere of small thickness that does not contain any of the poles 0 and π . Here the case is considered when the elasticity modules vary linearly with respect to the radius. It is assumed that the lateral surface of the sphere is free of stresses, and at the ends of the sphere (at the conical sections) the stresses are set, leaving it in equilibrium. A characteristic equation is obtained and, based on its asymptotic analysis, the existence of three groups of roots is established with respect to the small parameter characterizing the thickness of the sphere. The corresponding homogeneous solutions are constructed, depending on the roots of the characteristic equation. It is shown that the penetrating solution corresponds to the first group of roots. The second group of roots corresponds to the solution of the edge effect type, similar to the edge effect in the applied theory of shells. The third group of roots corresponds to the boundary layer type solution localized in the conical sections. The solution corresponding to the first and second groups of roots determines the internal stress–strain state of the sphere. In the first term of the asymptotic, they can be considered as a solution in the applied theory of shells. The question of satisfying the boundary conditions at the ends (on the conical sections) of the sphere is considered using the variational Lagrange principle.

中文翻译：

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具有可变弹性模量的球体弹性理论轴对称问题解的渐近行为

本文针对不包含任何极点 0 和 π 的小厚度径向非均匀球体研究了弹性理论的轴对称问题。这里考虑弹性模量相对于半径线性变化的情况。假定球体的侧表面没有应力，并且在球体的末端（在圆锥截面处）设置了应力，使其处于平衡状态。得到了一个特征方程，并在其渐近分析的基础上，针对表征球体厚度的小参数，建立了三组根的存在性。根据特征方程的根构造相应的齐次解。结果表明，穿透解对应于第一组根。第二组根对应于边缘效应类型的解，类似于壳的应用理论中的边缘效应。第三组根对应于锥形截面中的边界层类型解。第一组和第二组根对应的解决定了球体的内应力应变状态。在渐近的第一项中，它们可以被认为是壳的应用理论中的一个解。使用变分拉格朗日原理考虑满足球体端部（在圆锥截面上）边界条件的问题。第三组根对应于锥形截面中的边界层类型解。第一组和第二组根对应的解决定了球体的内应力应变状态。在渐近的第一项中，它们可以被认为是壳的应用理论中的一个解。使用变分拉格朗日原理考虑满足球体端部（在圆锥截面上）边界条件的问题。第三组根对应于锥形截面中的边界层类型解。第一组和第二组根对应的解决定了球体的内应力应变状态。在渐近的第一项中，它们可以被认为是壳的应用理论中的一个解。使用变分拉格朗日原理考虑满足球体端部（在圆锥截面上）边界条件的问题。